Let S={0, (1)/(4), (1)/(2), (3)/(4)}. (a) Show that the sample vectors corresponding to the func

step 1 Here, f1 is equal to 1, F2 is equal to sine x, F3 is equal to cosine x. So our B1 is equal to f1

Let S={0, (1)/(4), (1)/(2), (3)/(4)}. (a) Show that the...

Let $S=\left\{0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}\right\}$. (a) Show that the sample vectors corresponding to the functions $1, \cos \pi x, \cos 2 \pi x$, and $\cos 3 \pi x$ form a basis for the vector space of all sample functions on S. (b) Write the sampled version of the function $f(x)=x$ in terms of this basis.

Key Concepts

Linear Independence and Span

To confirm a set of functions forms a basis, one must verify they are linearly independent (no nontrivial combination of them equals the zero vector) and that they span the space (any vector in the space can be represented as a combination of these functions). These fundamental properties guarantee the set is a complete and non-redundant representation of the vector space.

Representation of Functions in a Given Basis

Once a basis is established, any function defined on the sample set can be uniquely represented as a linear combination of the basis functions. This involves determining coefficients such that the combination equals the function’s values at the sample points, effectively reconstructing the function from its basis expansion.

Finite-Dimensional Vector Space

In the context of sampled functions (or vectors), the collection of all function values corresponding to a finite set defines a finite-dimensional vector space. Each function is represented by a finite list (or vector) of its values at the sample points, and the dimension of the space equals the number of sample points.

Basis of a Vector Space

A basis of a vector space is a set of vectors that are linearly independent and span the entire space. In this context, showing that a set of sampled functions forms a basis means proving that every function defined on the sample set can be expressed as a unique linear combination of these basis functions.

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